Theoretical Analysis of Barrow Holographic Dark Energy in Fractal Cosmology

Abstract

We investigate interacting Barrow holographic dark energy (BHDE) with the Hubble horizon as the infrared cutoff in a fractal cosmological background. The fractal measure modifies the Friedmann sector and leads to the nonstandard closure relation Ωdm+Ωde=1+γ. Using Planck-inspired present-day normalization Ωde0=0.6847 and the benchmark parameters (Δ,ω,β)=(0.8,0.263,0.123), we study four linear and nonlinear dark-sector interactions over −0.99≤z≤3. The benchmark solutions remain within the physical background domain throughout this interval: the density fractions are non-negative, the relevant denominators remain positive, and the nonlinear Q3 and Q4 terms remain real. All four prescriptions exhibit a transition from decelerated to accelerated expansion. The non-interacting limit gives zt≃0.86, whereas for ξ=0.12 the transition redshifts are approximately 2.02, 1.97, 1.26, and 1.87 for Q1, Q2, Q3, and Q4, respectively; in particular, the Q4 solution recovers a finite transition within the plotted redshift range. A stronger positive coupling generally shifts acceleration onset to a higher redshift, with the strongest response for Q1 and the weakest for Q3. The statefinder quantities S3(1), S3(2), and the s–r trajectories distinguish the interaction structures through their finite-redshift evolution and present-day values. A fixed-background SN Ia comparison using 1046 selected Pantheon supernovae, diagonal FITRES uncertainties, and an analytically profiled additive nuisance parameter gives the lowest information criteria for flat ΛCDM. Among the BHDE benchmarks that interact, Q4 is the closest case, with ΔAIC=ΔBIC≃1.21. This comparison is not intended as a global posterior constraint on the model parameters.